Inequality Comparisons in a Multi-Period Framework:The Role of Alternative Welfare Metrics

John Creedy, Elin Halvorsen and Thor O. Thoresen

WORKING PAPER 08/2012September 2012

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Inequality Comparisons in a Multi-PeriodFramework: The Role of Alternative Welfare

Metrics∗

John Creedy†, Elin Halvorsen‡ and Thor O. Thoresen‡

Abstract

This paper considers the use of alternative welfare metrics in evaluations of in-come inequality in a multi-period context. Using Norwegian longitudinal incomedata, it is found, as in many studies, that inequality is lower when each individ-ual’s annual average income is used as welfare metric, compared with the use ofa single-period accounting framework. However, this result does not necessarilyhold when aversion to income fluctuations is introduced. Furthermore, when ac-tual incomes are replaced by expected incomes (conditional on an initial period),using a model of income dynamics, higher values of inequality over longer periodsare typically found, although comparisons depend on inequality and variabilityaversion parameters. The results are strongly influenced by the observed highdegree of systematic regression towards the (geometric) mean, combined with alarge extent of individual unexpected effects.

Keywords: income mobility, welfare evaluations, relative mobility processJEL: D31; C23; I31

∗This work was supported by the Research Council of Norway. We thank Philippe van Kerm andother participants at the microseminar of CEPS/INSTEAD, Differdange (Luxembourg), for commentson an earlier version of this paper. Useful comments were also obtained from John K. Dagsvik andparticipants at seminars in Statistics Norway.

†Victoria University of Wellington and New Zealand Treasury, on leave from The University ofMelbourne. E-mail: jcreedy@unimelb.edu.au

‡Research Department, Statistics Norway, 0033 Oslo, Norway. E-mail: elin.halvorsen@ssb.no,thor.olav.thoresen@ssb.no

1

1 Introduction

Evaluations of changes in the distribution of income must begin by deciding on a

number of fundamental ingredients, each of which involves value judgements. First, a

choice of ‘welfare metric’, concerning what is to be measured for each unit of analysis,

must be made. Secondly, a decision is needed regarding the time period of analysis.

Thirdly, the unit of analysis itself must be chosen. Finally, the form of ‘social eval-

uation function’, which encapsulates further explicit distributional value judgements,

has to be specified. The present paper explores the use of alternative welfare metrics

in a multi-period context, using the individual as the basic unit of analysis and an ad-

ditive, individualistic Paretean social welfare function reflecting belief in the ‘principle

of transfers’ (whereby a transfer from relatively rich to poor individuals, leaving their

rankings unchanged, is considered an improvement). The welfare metrics are based on

alternative income concepts rather than, say, consumption or utility measures which

allow for variations in the value of leisure time.

Consideration of a multi-period context necessarily introduces the role of income

mobility. This implies that inequality of income measured over a longer period is lower

than that in the highest single year.1 A further argument concerns comparative static

changes: if higher annual income inequality is associated with increased relative in-

come mobility, it is possible that inequality of income measured over several years is

lower. Hence, longer-period inequality may fall, and welfare might increase, despite the

rise in annual income inequality: this is referred to as a ‘mobility offsetting’ argument.

However, the welfare metric could allow for other effects.2 For example, if there is

imperfect substitutability of incomes over time (Atkinson, Bourguignon and Morris-

son, 1992) and individuals are averse to income variability, the offsetting argument is

weakened; see Creedy and Wilhelm (2002).

The discussion is typically, as above, carried out in terms of ex-post income mea-

sures. An alternative approach, explored here, is to attempt to allow explicitly for the

uncertainty associated with mobility by constructing a welfare metric based on an ex-

ante income measure. This in turn requires the use of a model of expectations formation

based on observed income dynamics. The association between mobility and ex-ante

1Conditions under which inequality is lower than in all years are examined by Creedy (1997a).2The question of whether income mobility represents equality of opportunity, as in Bénabou and

Ok (2001a), is not considered here.

2

income uncertainty has also been stressed by Parker and Rougier (2001), Gottschalk

and Spolaore (2002)3 and Ben-Shahar and Sulganik (2008).4

This paper presents results where expected income is derived by estimating an au-

toregressive model of income dynamics. A closed-form expression for expected income,

conditional on initial income, is obtained. Thus a ‘rational expectations’ approach is

used, whereby individuals are assumed to form expectations based on the dynamic

model of incomes and associated parameter estimates. The model specifies the log-

arithm of individual income in a given period as a function of the relative distance

from the geometric mean of a previous period’s income, an individual fixed effect and

a stochastic component. The social welfare function, and hence distributional value

judgements, examined are based on the Atkinson (1970) inequality index. To illustrate

the framework, longitudinal data for individuals in Norway over the period 1993—2005

are used.

An alternative approach to measuring long-period inequality involves the use of a

‘utility-equivalent annuity’, introduced by Nordhaus (1973) and defined as the constant

value which gives the same lifetime utility as the actual time profile.5 This can clearly

allow for an aversion to variability as well as imperfect capital markets. This concept

has recently been used by Aaberge and Mogstad (2010) and Aaberge et al. (2011)

to examine equality of opportunity.6 They distinguish two cases. First, equality of

opportunity is reflected in equal outcomes for all those with the same ‘effort’, so that

emphasis is on within-group inequality of utility-equivalent annuity. Second, between-

group comparisons are made where groups are instead defined by common ‘opportunity

sets’, and within-group inequality arising from different degrees of ‘effort’ are considered

irrelevant. The authors refer to the former as an ex-post approach and the latter as an

ex-ante approach. This interesting perspective clearly differs from the ex-ante concept

examined in the present paper.

3They present a decomposition analysis where the extent to which future incomes depend on currentincome is separated from effects due to rank reversals. For other decompositions, see Ruiz-Castillo(2004), Van Kerm (2004) and Jenkins and Van Kerm (2006).

4Other studies involving mobility and long-term incomes include, for example, Shorrocks (1978a),Chakravarty, Dutta and Weymark (1985), Fields (2010) and Hungerford (2011). For surveys of mo-bility, see Atkinson, Bourguignon and Morrisson (1992), Maasoumi (1998) and Fields and Ok (1999).

5Comparisons using this and alternative measures, using a lifetime simulation model, are reportedin Creedy (1997b).

6They also measure mobility in terms of the reduction in this longer-period income concept, fol-lowing Shorrocks (1978a).

3

In Section 2 the data and the Atkinson index are briefly described. Section 3

presents results using ex-post welfare metrics. Section 4 introduces ex-ante income

uncertainty and presents a procedure for using expected future incomes in the welfare

metric. Section 5 summarises the main findings.

2 Data and Inequality Measurement

The data used below come from Income Statistics for Persons and Families in Norway

1993—2005 (Statistics Norway, 2006). These data contain register-based information

on the whole population, derived primarily from information retrieved from all income

tax returns in the Directorate of Taxes’ Register of Personal Tax-Payers. The choice

of time period, 1993—2005, is conditioned on the register being established in 1993,

and the desirability of avoiding the tax reform of 2006, as a reform normally involves

measurement challenges. Nevertheless, as data primarily are used to illustrate the

alternative metrics, the choice of time period mainly follows from the desire to explore

data from a sufficiently long time period to be divided into two periods of equal length.

The income measure is annual income after tax. Thus income is defined as labour

income, plus positive capital income, plus net capital gains, plus transfers minus direct

taxes. This is the definition used in all official income statistics in Norway. Negative

capital income (interest paid on mortgages) is not included in the definition because

there is no corresponding income from housing in the statistics. Estimates of income

mobility are typically sensitive to persons entering and leaving the labour market.

Hence, persons under age 26 and above 65 are excluded, and those with an income

below 100NOK7 in any year are excluded. The effects of inflation have been removed

by deflating all incomes to the 1998 level using the consumer price index.

Results are presented using the well-known Atkinson (1970) inequality measure.

Let individual i’s income (the welfare metric, ignoring time for now) be denoted yi,

for i = 1, ..., n. The Atkinson measure is based on the additive social welfare function,

W = W (y1, ..., yn) of the form W =∑n

i=1 U (yi), where U (yi) is the weight attached

7This is equivalent to US$15.50 using 2005 exchange rate.

4

to yi, and is specified, for ε �= 1, as:8

U (yi| ε) =y1−εi

1− ε(1)

Hence ε ≥ 0 captures the concavity of U , corresponding to the aversion to relative

inequality. Let yEDE denote the equally distributed equivalent income, that is, the

income which, if obtained by each person, gives the same social welfare as the actual

distribution. Hence, for ε �= 1:

yEDE =

(1

n

n∑i=1

y1−εi

) 1

1−ε

(2)

Atkinson’s index of inequality, I, is the proportional difference between the arithmetic

mean, y, and yEDE, so that:

I =y − yEDE

y(3)

and I reflects the ‘wastefulness of inequality’.

3 Alternative Ex-post Evaluations

Figures 1 and 2 show, for the period 1993—2005 and for two inequality aversion pa-

rameters, the time profiles of inequality and the equally distributed equivalent. The

period may be divided into two periods of equal length, 1994—99 and 2000—05.9 The

first period reflects a relatively stable degree of inequality while the second period

displays more variability, initially decreasing and then increasing steadily. Both the

general economic development and tax-payers’ behavioural reactions to tax legislative

changes have influenced the observed income patterns. The first part of the time pe-

riod coincides with a period of high economic growth (GDP growth above 4 per cent

in the period 1994—97), then growth rates are lower and more variable in the period

1998—2005 (but above 2 per cent in all years, except in 2003). However, from an income

distribution perspective, it is suggested that the development of the personal income

tax schedule is at least equally important. The tax reform of 1992 introduced a dual

8If ε = 1, y1−εi/ (1− ε) is replaced by log y.

91993 is therefore not used in inequality comparisons, but as a base year when estimating ex-anteincomes.

5

income tax system, which combines a low proportional tax rate on capital income and

progressive tax rates on labour income.

These separate schedules for capital and labour income created obvious incentives

for taxpayers to take advantage of the lower tax on capital. For example, Thoresen

and Alstadsæter (2010) show that owners of small businesses were able to gain from

this schedule by changing organizational form. Overall, there was a notably increase

in (low taxed) dividend income transfers to households over the period, from less than

10 billion in 1994 to nearly NOK100 billion in 2005. Only 2001 does not fit into the

steady upward trend, as there was a temporary tax on dividends for shareholders that

year. Furthermore, corporations brought forward their distribution of profits because

of the pre-announced tax reform of 2006 (Alstadsæter and Fjærli, 2009), given that

it was evident that the reform introduced taxation of dividends at both the corporate

and individual level (in contrast to the 1992 reform, which had only corporate-level

taxation), through a shareholder income tax; see Sørensen (2005) for further details.

As dividend income is unequally dispersed — for example, 95 per cent of dividends were

received by individuals in decile 10 in 2004 — this has resulted in a substantial increase

in post-tax income inequality over the period (with an exception for 2001) and changes

in the composition of income across income distributions; for more details see Lambert

and Thoresen (2009, Figure 3).

A time subscript must now be added to each individuals’ income. For convenience,

the following ignores discounting. Consider first an ex-post evaluation over T periods

which uses as welfare metric for each individual the average annual income, yi =1T

∑T

t=1 yit. Hence the welfare function is not actually concerned with the way in which

any individual’s income is distributed over the time period, and thus may be said to

reflect a lack of concern for the nature of the mobility process. For the period 1994—

2005, the use of the average annual income as welfare metric for each individual gives

Atkinson inequality measures of 0.076, 0.134 and 0.210 respectively for values of ε of

0.5, 1.0 and 1.5. These each reflect value judgements which tolerate substantial leaks in

making equalising transfers.10 These values may be compared with the annual average

inequality measures of 0.099, 0.181 and 0.298 respectively. The use of a longer period

whereby individual incomes are averaged is thus equalising in this case.

10For example, if 1 unit is taken from A to make a transfer to B, where A is twice as rich as B, atransfer of 0.5 units leaves social welfare unchanged if ε = 1. This falls to 0.25 units if ε = 2.

6

0.100

0.120

0.140

0.160

0.180

0.200

0.220

0.240

1993 1995 1997 1999 2001 2003 2005

0.000

0.020

0.040

0.060

0.080

0.100

0.120

0.140

0.160

A(1), left axis A(0.5), right axis

Figure 1: Atkinson Inequality Measures, Annual Income 1993-2005, ε = 0.5 and ε = 1

120,000

130,000

140,000

150,000

160,000

170,000

180,000

190,000

200,000

210,000

220,000

1993 1995 1997 1999 2001 2003 2005

Y_EDE(0.5) Y_EDE(1)

Figure 2: Equally Distributed Equivalent Annual Income: 1993-2005

7

Table 1: Atkinson Inequality Measures: Annual Average Income Inequality and In-equality with Annual Average as Welfare Metric

Period 1994—1999 Period 2000—2005ε = 0.5 ε = 1.0 ε = 1.5 ε = 0.5 ε = 1.0 ε = 1.5

Inequality: Annual average of single year valuesIy 0.084 0.165 0.263 0.108 0.188 0.307

Using each individual’s annual average as welfare metricIy 0.071 0.134 0.183 0.082 0.147 0.228

Further details for the two sub-periods are shown in Table 1.11 For the second

period, the absolute reduction in inequality when using annual average income as the

welfare metric compared with annual average inequality, is double the reduction ob-

tained for the first period. The inequality-reducing effects of using a longer accounting

period, mentioned above, therefore appears to be greater in a period when annual

inequality is generally increasing.12 However,the percentage reduction in inequality

is larger in the first period for ε = 1.5. This arises because the very high degree of

inequality aversion places more emphasis on the low end of the income distribution.

The income mobility which produces the increasing annual inequality is thus also

responsible for reducing the inequality of a multi-period income measure (each person’s

annual average) below average annual inequality. However, mobility may not necessar-

ily be seen as beneficial from an individual’s point of view. It may also be seen as an

undesirable source of economic instability. For example, individuals may for some rea-

son be unable to smooth consumption over time when facing income fluctuations and

they might be averse to such variability in income. Imperfections of capital markets or

other constraints may prevent individuals from smoothing consumption over time; see

Atkinson, Bourguignon and Morrison (1992).13

11Standard errors of the estimates, which can be obtained by bootstrapping procedures, are notreported as they in general are very small due to the sample size (more than 2,600,000 observations).

12This of course differs from the comparative static argument discussed in the introduction, andexamined in detail by Creedy and Wilhelm (2002) in terms of inequality and social welfare.

13Shorrocks (1978a) argues that mobility is always desirable, whereas Chakravarty, Dutta and Wey-mark (1985) establish a no-mobility hypothetical benchmark from which they can distinguish betweendesirable and undesirable mobility. Like King (1983), they make use of the equally distributed equiva-lent idea. The difficulties of establishing a reasonable social welfare understanding of income mobilityis discussed by Atkinson (1981), Dardanoni (1993) and Fields (2010).

8

It may therefore be desired to allow, in the welfare metric, for an aversion to income

variability, as suggested by Creedy and Wilhelm (2002); see also Jarvis and Jenkins

(1998) on the disutility of income volatility. This can be done, in an ex-post context,

by using instead of yi a welfare metric, yi, defined as:

yi =1

T

T∑t=1

y1−γit

1− γ(4)

The parameter γ measures the degree of aversion to variability of income over time, and

the same parameter is assumed to apply to all individuals. As ex-post values are used,

the aversion coefficient, γ, is not interpreted in terms of risk aversion: this is discussed

in Section 4 below. The relative values of ε and γ determine whether inequality aversion

(of the judge whose value judgements are represented by the welfare function) is high

enough to overcome the individuals’ aversion to income variability over time. When

aversion to income variability is high relative to inequality aversion, a more ‘static’

society is preferred, in which income is more stable at the ‘cost’ of higher inequality of

multi-period income.

Table 2: Atkinson Inequality Measures with Aversion to Income Fluctuations

Period 1994—1999 Period 2000—2005ε = 0.5 ε = 1.0 ε = 1.5 ε = 0.5 ε = 1.0 ε = 1.5

γ = 0 0.071 0.134 0.217 0.082 0.147 0.289γ = 0.5 0.085 0.148 0.236 0.098 0.161 0.311γ = 2.0 0.130 0.206 0.349 0.144 0.217 0.375γ = 3.0 0.151 0.235 0.427 0.166 0.244 0.499

Table 2 shows the extent to which the values in Table 1 are increased when aversion

to intertemporal fluctuations is introduced. In order to eliminate the effect of general

income growth over time, incomes were adjusted so that average annual income is

constant (and equal to the overall mean) in each period. Hence the inequality values in

the first row of Table 2 differ slightly from those in the final row of Table 1. It is clear

that inequality increases as individuals’ aversion to income variability increases. The

inequality differences between the first and second sub-periods are less influenced by

aversion to income variability over time. The differences for γ = 0 and positive γs are

approximately similar in the two subperiods. As the last period involves a temporary

tax on dividends for the shareholders in 2001, one may expect stronger effects from

9

increases in the value of γ in that period, but no clear manifestation of such effects is

evident in Table 2.

4 An Ex-ante Perspective

The suggestion that relative income mobility is associated with uncertainty leads to the

idea that an alternative evaluation may be based on an ex-ante measure, rather than ex-

post incomes as in the previous section. For example, Shorrocks (1978b, p.1016) argues

that ‘interest in mobility is not only concerned with movement but also predictability’.

Furthermore, the uncertainty aspect of mobility is emphasised by contributions which

see mobility in terms of future opportunities, as in Bénabou and Ok (2001), or account

for origin independence, as in Gottschalk and Spolaore (2002). An ex-ante perspective

is introduced in subsection 4.1. The approach requires a model of income dynamics

and this is described in subsection 4.2. Results using the new metric are presented in

subsection 4.3

4.1 The Welfare Metric

The approach considered here is to replace the above welfare metric with one defined

in terms of expected incomes, conditional on income in a specified period, E (yit| yi0),

so that (4) is replaced by:

E (yi) =1

T

T∑t=1

E(y1−γit

∣∣ yi0)

1− γ(5)

Here the parameter γ can be interpreted in terms of risk aversion. In a one-commodity

setting and with indifference with respect to the timing of risk, risk aversion is the

inverse of the elasticity of intertemporal substitution. Thus resistance to intertemporal

substitution, or variability aversion, is closely related to risk aversion.

Application of this approach therefore requires knowledge of the conditional expec-

tation of future incomes. The following subsection proposes a measure of expected

income obtained by modelling the income process.

10

4.2 Modelling Income Dynamics

The aim of this section is to present and estimate a simple model of individuals’ ex-

pectations of future incomes which can be used to produce ex ante income measures.

Consider a dynamic process containing both a stochastic component and a component

in which changes depend on the position of individuals relative to the geometric mean;

see also Creedy (1985) and Creedy and Wilhelm (2002). As before, yit denotes individ-

ual i’s income in period t, and let µt denote the mean of logarithms in period t, with

mt = exp (µt) as the geometric mean. The income process can be written as:

yit =

(yit−1mt−1

)βexp (µt + vi + ηit) (6)

where the stochastic component consists of an individual-specific effect, vi, and a ran-

dom component, ηit, assumed to be independent of income, with zero mean and a

variance in each period of σ2η. Equation (6) can be rewritten as:

(log yit − µt) = β(log yit−1 − µt−1

)+ vi + ηit (7)

The autoregressive parameter, β, captures variations in income which decline more

slowly over time. In other words it reflects movements in income that, while not per-

manent, tend to persist for several years. Suppose also that in this simple income

process, the autoregressive parameter and income variance is common for all individu-

als, and heterogeneity in the process is represented through the individual fixed effect

(individual fixed level relative to the mean) and the error term.

Table 3 reports results of using several estimators. These include the least squares

dummy variables (LSDV), and generalised method of moments (GMM) estimators

as in Arellano and Bond (1991) and Blundell and Bond (1998). Because the lagged

dependent variable is correlated with the error term, it has been shown that the use of

LSDV result in biased estimates. Anderson and Hsiao (1981) suggested first eliminating

the fixed effect by taking first differences, and then using yt−2 as instrument for ∆yt.

However, this does not exploit all the relevant moment conditions so it is not the

efficient GMM estimator. Arrelano and Bond (1991) derived other moment conditions

to be used in GMM estimation. This estimator is known as the Arrelano-Bond GMM

estimator. Other instruments have been suggested by a succession of researchers, such

as the Arellano and Bover/Blundell and Bond system estimator (Arellano and Bover,

11

Table 3: Parameter Estimates for the Income Mobility Process

Method and Parameter All years 1994-1999 2000-2005LSDVβ 0.452 0.279 0.227

(0.001) (0.001) (0.001)ση 0.276 0.280 0.282AB-GMMβ 0.492 0.477 0.351

(0.002) (0.001) (0.002)ση 0.415 0.308 0.316GMMSYSβ 0.476 0.486 0.387

(0.001) (0.001) (0.001)ση 0.419 0.309 0.313GMM-MA(1)β 0.473 0.527 0.443

(0.002) (0.003) (0.003)ση 0.524 0.406 0.436Note: Robust standard errors are in parentheses below parameter estimates.

1995; Blundell and Bond, 1998), which uses moment conditions in which lagged first

differences of the dependent variable are instruments for the level equation. In practice,

it is difficult to find good instruments for the first-differenced lagged dependent variable,

which can itself create problems for the estimation. Kiviet (1995) shows that panel

data models using instrumental variable estimation often lead to poor finite sample

efficiency and bias. Also, tests show that none of the methods rejects the assumption

of no autocorrelation in first differenced errors. Thus, the final specification is a GMM

model assuming moving-average serial correlation in the residuals.14

Table 3 shows that a common result for all specifications is that the estimated value

of β is higher in the first sub-period than in the second sub-period, while the estimate

of σ2η is higher in the second sub-period. Since the standard errors are low, it may

be inferred that the estimated β’s in the two periods are also significantly different

from each other. The lower degree of regression towards the mean and lower variance

in the first period implies lower income mobility, and therefore higher predictability

14Lags three or higher are used as valid instruments for the differenced equation.

12

of future incomes. Conversely, the parameter values imply higher mobility and less

predictability in the second sub-period. In the following subsection, reported results

are based on the Arrelano-Bond GMM estimator.

The model specified in (7) is simple compared with a number other approaches

used. A less parsimonious model, such as the error component model, which is now a

standard model in the income process literature15 would probably provide more reliable

estimates for the income process. Adding heterogeneity in the model parameters as

in Baker and Solon (2003) and Browning et al. (2010) would improve the model even

more.

However, the basic model used here has been chosen for two reasons. First, it is

helpful to keep the perspective of the social planner. An income process is specified that

the planner is assumed to use for prediction of future incomes, based on observations

of current incomes. All individual characteristics, observable or unobservable, that

may explain individual income levels are captured by the individual fixed effect. We

assume that the planner, possibly with the help of an econometrician, has estimated

the distribution of these fixed effects on historical data, and furthermore, assumes

that these are constant over time. Second, the simple autoregressive income process

is in line with the Markov-models often used in the income mobility index literature,

and therefore provides a link between the income mobility literature and the more

econometrically orientated income process literature.

Including other explanatory variables, such as age, family composition and edu-

cation would substantially complicate the prediction of future incomes. While age is

straightforward to predict, prediction of future family composition is rather demand-

ing. Education is challenging too, as the specification already accounts for a fixed

effect. Thus, fixed effects soak up much of the explanatory power of variables that

are either time-invariant or close to time-invariance. We have explored the effects of

using other explanatory variables, such as age, family composition and education. The

estimated autoregression coefficients became somewhat lower, but the overall result

did not change. This suggests that the main difference between income mobility in the

two periods is due to genuine income dynamics rather than, for instance, substantial

differences in family dynamics. Comparisons were also made using alternative income

15This literature is represented by the works of Lillard and Willis (1978), MaCurdy (1982), Abowdand Card (1989), Baker (1997), Carroll and Samwick (1997), and Ramos (2003).

13

definitions. Labour income yields similar estimates for the autoregression coefficient,

but exhibits a much larger variance. For gross income there is less regression towards

the mean (that is, higher β) than for the two other income definitions, and the differ-

ence between the two periods is larger. Also, as expected, the standard deviation of

gross income is higher than for income after tax.

4.3 Inequality Using the Ex-ante Welfare Metric

In order to obtain measures for the contribution of the estimated income process to

the overall ex-ante welfare evaluation, a closed-form expression for expected income as

a function of income in the initial period, E (yit|yi0), is required. It is shown in the

Appendix that:

E (yit|yi0, vi) = exp

{µt + β

t (log yi0 − µ0) + vi

(1− βt

1− β

)+

1− β2t

2(1− β2

)σ2η

}(8)

where estimates of individual fixed-effects, vi, are obtained using their sample counter-

parts. The corresponding equally distributed equivalent in terms of expected income,

EDEE(y|y0), can be expressed as:

EDEE(y|y0) =

[1

n

n∑i=1

(1

T

T∑t=1

E (yit|yi0)1−γ

) 1−ε

1−γ

] 1

1−ε

(9)

from which, given the arithmetic mean, the Atkinson measure can be obtained in the

usual way. In this case it depends on the degree of regression towards the mean, the

income variance, the degree of aversion to inequality and the degree of aversion to

fluctuations in income. When β < 1, the initial (relative) position is given less weight

over time, while the role of the individual-specific position is increasing over time.

Expected income is also increasing over time.

The inequality measures for the ex-ante welfare metric are shown in Table 4, where

again any effects of income growth are eliminated by maintaining arithmetic mean

constant. These may be compared with Table 2. For the sub-period 1994—99, inequality

is lower for all values of ε examined and for the variability aversion coefficients of 0 and

0.5. For the very high values of γ of 2.0 and 3.0, inequality is higher when the ex-ante

measure is used, particularly for the high inequality aversion coefficient. The estimated

value of β is rather low while that of σ2η is high compared with values reported in earlier

14

Table 4: Atkinson Inequality Measures for Expected Income

Period 1994—1999 Period 2000—2005ε = 0.5 ε = 1.0 ε = 1.5 ε = 0.5 ε = 1.0 ε = 1.5

γ = 0 0.004 0.068 0.135 0.109 0.165 0.218γ = 0.5 0.037 0.095 0.165 0.150 0.208 0.246γ = 2.0 0.132 0.221 0.417 0.250 0.367 0.492γ = 3.0 0.166 0.394 0.513 0.290 0.481 0.603

studies; see Creedy (1985). The considerable variability implied by the high σ2η would

produce increasing annual inequality over time, without the low value of β, implying

considerable regression towards the mean. In the expression for E (yit|yi0), the effects

of terms involving powers of β rapidly become insignificant. Expected incomes are

dominated by the high σ2η which, for the high mobility-aversion cases, implies a higher

measured inequality. From (8), setting all terms involving βt and β2t to zero16 and

rearranging gives:

logE (yit)− µt =vi

1− β+

σ2η

2(1− β2

) (10)

Hence the variance of logarithms of expected income soon becomes σ2v/ (1− β)2, where

σ2v is the variance of the fixed effect in the autoregressive income-generation equation.

Therefore for higher σ2v and lower β, as in the second sub-period considered, inequality

of expected values is quickly increasing towards a relatively high value.

In the ex-post case, there is less inequality than anticipated as a result of the re-

gression towards the mean. For the second sub-period, the role of unanticipated, but

systematically equalising, mobility is even greater and β is lower. Hence the ex-ante

welfare metric produces higher inequality, for nearly all combinations of variability

aversion and inequality aversion parameters, than for the ex-post metric. The ex-

ceptions are for the combination of low variability aversion with very high inequality

aversion. Also, the inequality differences between the two sub-periods are maintained

or increased when moving from the ex-post to the ex-ante perspective.

Discounting has been ignored here for ease of exposition. Introducing discounting

of time periods would imply that greater weight is placed on the first period. In other

words, the initial distribution would play a relatively larger role than without discount-

16This is the same as replacing E (yit|yi0) by E (yit), that is the unconditional expectation.

15

ing. As long as time preference is homogeneous across individuals and constant over

time, introducing discounting would not change the qualitative difference between the

two sub-periods considered.17 But, as the initial distributions become more dominant

under discounting, the inequality estimates in Table 4 would be modestly increased.

It is therefore necessary to consider the role of the initial distributions. It may be

argued that it is difficult to interpret the results for the two periods in Table 4 in terms

of different income processes because they begin with different initial distributions.

For this reason two sensitivity analyses were carried out. First, it was assumed simply

that (log yi0 − µ0) = 0 for all individuals (so that the fixed effect is the only individual

variation). Second, the same initial distribution was used in both periods (hence, the

second period process was estimated using the initial 1993 distribution). Unreported

results show that the results are not sensitive to the choice of initial distributions. This

lack of sensitivity is likely to arise because of the high degree of regression towards the

mean.

5 Conclusions

The aim of this paper has been to consider the use of alternative welfare metrics in eval-

uations of income inequality when a multi-period income measure is used, and hence

relative income mobility plays a crucial role in influencing the relationship between

short- and long-period inequality. One basic approach, most commonly adopted in in-

come distribution studies, is to base measures on ex-post magnitudes. Using Norwegian

longitudinal income data, it was found, as in many studies, that income inequality is

lower when each individual’s annual average income is used as welfare metric, compared

with the use of a single-period accounting framework. However, the longer accounting

period can produce both lower and higher inequality than annual measures, depending

on the assumed degree of aversion to income fluctuations over time.

The second approach took as its starting point the argument that relative income

mobility introduces uncertainty about future incomes, so that it may be desired to

evaluate inequality using an ex-ante approach. To this end, a regression model of in-

come dynamics was used in order to generate individuals’ expected values of future

17However, if individuals were not indifferent to the timing of risk, then introducing discountingwould lead to a preference for early resolution of risk.

16

income, conditional on actual income in a specified initial period. The use of expected

incomes was found generally to produce higher values of inequality over longer peri-

ods, although again comparisons depend on the assumption regarding the aversion to

income inequality of the social welfare function, and aversion to income fluctuations

on the part of individuals. The results were strongly influenced by the observed high

degree of systematic regression towards the (geometric) mean, combined with a large

extent of random proportional income changes. The distinction between expected and

unexpected mobility was thus found to be important.

In the choice of welfare metric there is of course no single ‘correct’ approach, and the

contribution of the economist is to investigate the implications of adopting alternative

value judgements. The present paper is therefore in this spirit of extending the range

of value judgements which can be examined.

17

Appendix A: Expected Income

This appendix derives the expected value of an individual’s income, conditional on

income in an earlier period. Define zt, dropping individual subscripts, as:

zt = (log yt − µt) (A.1)

Inserting zt in equation (7) gives:

zt = βzt−1 + v + ηt (A.2)

Backwards induction yields:

zt = v + ηt + β(v + ηt−1

)+ β2

(v + ηt−2

)+ ...+ βk−1

(v + ηt−k+1

)+ βtzt−k

= βtzt−k +k−1∑s=0

vβs + ξt (A.3)

where:

ξt = ηt + βηt−1 + β2ηt−2 + ...+ β

k−1ηt−k+1 (A.4)

Using yt = exp (zt + µt) from (A.1):

yt = exp

{µt + v

(1− βk

1− β

)+ ξt + β

t(log yt−k − µt−k

)}(A.5)

The variance of log yt is thus equal to V ar (ξt), which, if η is normally distributed,

is from (A.4) given by a weighted sum of normal variables. This is also normally

distributed, with V ar (∑

i aiXi) =∑

i a2iV arXi. Hence the variance of log yt is:

V ar

(k−1∑s=0

vβs)= σ2η

k−1∑

s=0

β2s =1− β2k

1− β2σ2η (A.6)

In general, if the variable x is lognormally distributed with mean and variance of

logarithms of m and s2 respectively, then E (x) = exp(m+ s2

2

). Taking expectations

of y, gives:

E (yt| yt−k) = exp

{µt + β

k(log yt−k − µt−k

)+ v

(1− βk

1− β

)+

1− β2k

2(1− β2

)σ2η

}(A.7)

Setting k = t gives the result in (8) above.

18

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22

Working Papers in Public Finance Series

WP01/2012 Debasis Bandyopadhyay, Robert Barro, Jeremy Couchman, Norman Gemmell,

Gordon Liao and Fiona McAlister, "Average Marginal Income Tax Rates in New Zealand,

1907-2009"

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New Zealand"

WP03/2012 Simon Carey, John Creedy, Norman Gemmell and Josh Teng "Regression

Estimates of the Elasticity of Taxable Income and the Choice of Instrument"

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Rate Changes in Multi-Rate Income Tax Systems: Behavioural and Structural Factors"

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Welfare Changes: The Use of Alternative Welfare Metrics"

WP07/2012 John Creedy and Solmaz Moslehi, "The Composition of Government

Expenditure with Alternative Choice Mechanisms"

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in a Multi-Period Framework: The Role of Alternative Welfare Metrics"

WP09/2012 Norman Gemmell and John Hasseldine, "The Tax Gap: A Methodological

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Working Papers in Public FinanceChair in Public Finance Victoria Business School

About the Authors John Creedy is Visiting Professorial Fellow at Victoria Business School, Victoria University of Wellington, New Zealand, and a Principal Advisorat the New Zealand Treasury. He is on leave from the University of Melbournewhere he is The Truby Williams Professor of Economics. Email: john.creedy@vuw.ac.nz Elin Halvorsen is a Research Economist at the Research Department, Statistics Norway. Email: e.halvorsen@ssb.no Thor O. Thoresen is a Senior Research Fellow at the Research Department, Statistics Norway. Email: thor.olav.thoresen@ssb.no